We study the eigenvalue distribution and resolvent of a Kronecker-product random matrix model $A \otimes I_{n \times n}+I_{n \times n} \otimes B+\Theta \otimes \Xi \in \mathbb{C}^{n^2 \times n^2}$, where $A,B$ are independent Wigner matrices and $\Theta,\Xi$ are deterministic and diagonal. For fixed spectral arguments, we establish a quantitative approximation for the Stieltjes transform by that of an approximating free operator, and a diagonal deterministic equivalent approximation for the resolvent. We further obtain sharp estimates in operator norm for the $n \times n$ resolvent blocks, and show that off-diagonal resolvent entries fall on two differing scales of $n^{-1/2}$ and $n^{-1}$ depending on their locations in the Kronecker structure. Our study is motivated by consideration of a matrix-valued least-squares optimization problem $\min_{X \in \mathbb{R}^{n \times n}} \frac{1}{2}\|XA+BX\|_F^2+\frac{1}{2}\sum_{ij} \xi_i\theta_j x_{ij}^2$ subject to a linear constraint. For random instances of this problem defined by Wigner inputs $A,B$, our analyses imply an asymptotic characterization of the minimizer $X$ and its associated minimum objective value as $n \to \infty$.

翻译：我们研究克罗内克积随机矩阵模型 $A \otimes I_{n \times n}+I_{n \times n} \otimes B+\Theta \otimes \Xi \in \mathbb{C}^{n^2 \times n^2}$ 的特征值分布与预解式，其中 $A,B$ 为独立的维格纳矩阵，$\Theta,\Xi$ 为确定性对角矩阵。对于固定谱参数，我们建立了斯蒂尔杰斯变换通过近似自由算子变换的定量逼近，以及预解式的对角确定性等价逼近。我们进一步获得了 $n \times n$ 预解块在算子范数下的锐利估计，并证明非对角预解元项根据其在克罗内克结构中的位置分别落在 $n^{-1/2}$ 和 $n^{-1}$ 两个不同尺度上。本研究受矩阵值最小二乘优化问题 $\min_{X \in \mathbb{R}^{n \times n}} \frac{1}{2}\|XA+BX\|_F^2+\frac{1}{2}\sum_{ij} \xi_i\theta_j x_{ij}^2$ 在线性约束下的考量所驱动。对于由维格纳输入 $A,B$ 定义的该问题随机实例，我们的分析暗示了当 $n \to \infty$ 时最小化子 $X$ 及其关联最小目标值的渐近特征刻画。